Resolution: 1920×1080 | Inclination: 45.0° | Spin: 0.90 | Orders: 3 | r ∈ [2.3, 20.0]M
Distribution of geodesic orders per pixel. Multiple orders indicate strong gravitational lensing.
| Bucket | Pixels | Share | Visual |
|---|---|---|---|
| Only order 0 (primary) | 273992 | 13.2% | |
| Orders 0 & 1 (+ ring) | 33148 | 1.6% | |
| Orders ≥2 (+ subrings) | 5887 | 0.3% | |
| No hit (miss) | 1760573 | 84.9% |
Classification of rays that don't hit the accretion disc.
| Miss Type | Count | Share |
|---|---|---|
| Escaped (r → ∞) | 1682449 | 95.6% |
| Captured (r → rh) | 78124 | 4.4% |
| Aborted (numerical) | 0 | 0.0% |
Escaped: Ray reaches r > 1000M (escapes to infinity)
Captured: Ray falls into event horizon (r < rh + 0.01M)
Aborted: Integration stopped due to numerical issues or step limit
Visual spatial distribution of geodesic orders. White pixels indicate hits at that order.
Primary disc image
Photon ring
Higher-order subrings
Top 10 rays with highest null invariant errors for quality inspection. These outliers warrant review but may still be within acceptable tolerances for complex geodesics.
| Rank | Pixel (x, y) | Order | NI Error | Attention Level |
|---|---|---|---|---|
| 1 | (660, 671) | 0 | 8.40e-9 | High Attention |
| 2 | (1060, 327) | 1 | 7.17e-9 | High Attention |
| 3 | (1060, 327) | 2 | 7.17e-9 | High Attention |
| 4 | (1275, 617) | 1 | 5.44e-9 | High Attention |
| 5 | (1275, 617) | 2 | 5.44e-9 | High Attention |
| 6 | (783, 608) | 0 | 4.77e-9 | High Attention |
| 7 | (1274, 762) | 1 | 4.48e-9 | High Attention |
| 8 | (841, 571) | 0 | 3.29e-9 | High Attention |
| 9 | (1047, 435) | 1 | 3.26e-9 | High Attention |
| 10 | (1028, 283) | 1 | 3.22e-9 | High Attention |
Low: NI < 1e-12 | Medium: 1e-12 ≤ NI < 1e-9 | High Attention: NI ≥ 1e-9
The Carter constant K = Q + (Lz - aE)² must be ≥ 0 for physical geodesics. Violations indicate integration errors.
2D histograms showing how image position maps to disc emission radius for each geodesic order.
Order 0 (Primary)
Order 1 (Photon Ring)
Order 2+ (Subrings)
Heatmap showing arrival time delays across the image via affine parameter λ. Warm colors indicate longer light paths.
Null invariant (NI) measures how well geodesics maintain the constraint gμν kμ kν = 0.
| Metric | Value |
|---|---|
| NI min | 0.00e0 |
| NI median | 0.00e0 |
| NI mean | 6.82e-13 |
| NI p95 | 1.56e-12 |
| NI p99 | 6.92e-12 |
| NI max | 8.40e-9 |
Good: Most rays are publishable quality, some numerical drift
Where disc hits occur. Peak near ISCO indicates brightest emission region.
| Radius Range | Hits | Share | Visual |
|---|---|---|---|
| [2.3-4.0M] Near ISCO | 7590 | 2.4% | |
| [4.0-8.0M] Inner disc | 33453 | 10.7% | |
| [8.0-14.0M] Mid disc | 97947 | 31.3% | |
| [14.0-20.0M] Outer disc | 174037 | 55.6% |
φ-wraps are unitless (2π cycles), λ is affine parameter at disc hit
| Order | Count | Mean φ-wraps | Mean λ |
|---|---|---|---|
| 0 (Primary) | 313027 | 0.59 | 63.6M |
| 1 (Ring) | 39035 | 0.42 | 102.5M |
| 2+ (Subrings) | 5887 | 0.46 | 140.8M |
Redshift factor g = νobs/νemit (unitless). Values >1 indicate Doppler boosting (approaching), <1 indicate dimming (receding).
Clear Doppler asymmetry from frame-dragging
φ-wraps measure how many times photons wind around the black hole before hitting the disc.
| Order | φ-wraps Range | Mean |
|---|---|---|
| 0 (Primary) | 0.00 - 1.04 | 0.59 |
| 1 (Ring) | 0.42 wraps (avg) | |
| 2+ (Subrings) | 0.46 wraps (avg) | |
Max wraps: 1.04 (moderate lensing (typical for disc imaging))
Extracted 109172 boundary pixels between captured and escaped rays.
| Parameter | Value |
|---|---|
| Center (x, y) | (895.8, 540.2) |
| Semi-major axis | 843.4 px |
| Semi-minor axis | 480.9 px |
| Rotation | 0.24° |
| Axis ratio (b/a) | 0.570 |
Fitted ellipse approximates the black hole shadow boundary. Axis ratio near 1.0 indicates circular shadow (low spin or face-on view).
Histogram of radial (r) and polar (θ) turning points. High counts indicate complex, chaotic geodesics near the photon sphere.
Relationship between impact parameter b = Lz/E and azimuthal wraps. The vertical red line marks the photon sphere radius where rays orbit multiple times before hitting the disc.
X-axis (Impact Parameter b): Angular momentum per unit energy (Lz/E). Negative values mean counter-rotation relative to black hole spin.
Y-axis (φ-wraps): Number of times the light ray winds around the black hole before hitting the disc.
Colors: Green = Order 0 (primary), Blue = Order 1 (photon ring), Purple = Order 2+ (higher subrings)
Photon Sphere (red line): Critical impact parameter (~3.79M for a=0.9) where photons can orbit the black hole. Rays near this value show the highest wraps.
Expected Pattern: A peak in wraps near bphoton, with Order 1 and 2+ rays concentrated in this region due to multiple orbits before disc intersection.
Comprehensive 2-pass outlier detection using robust statistical methods (MAD-based z-scores, Mahalanobis distance, spatial discontinuity analysis).
| Pixel | Order | Category | Value | Details |
|---|---|---|---|---|
| (560, 359) | 0 | Large Null Invariant Error | 1.002e-9 | NI=1.002e-9 (integration accuracy issue) |
| (660, 671) | 0 | Large Null Invariant Error | 8.399e-9 | NI=8.399e-9 (integration accuracy issue) |
| (703, 715) | 0 | Large Null Invariant Error | 1.025e-9 | NI=1.025e-9 (integration accuracy issue) |
| (763, 281) | 0 | Large Null Invariant Error | 1.025e-9 | NI=1.025e-9 (integration accuracy issue) |
| (770, 682) | 0 | Large Null Invariant Error | 1.797e-9 | NI=1.797e-9 (integration accuracy issue) |
| (780, 612) | 0 | Large Null Invariant Error | 1.597e-9 | NI=1.597e-9 (integration accuracy issue) |
| (783, 608) | 0 | Large Null Invariant Error | 4.766e-9 | NI=4.766e-9 (integration accuracy issue) |
| (819, 421) | 0 | Large Null Invariant Error | 1.127e-9 | NI=1.127e-9 (integration accuracy issue) |
| (819, 421) | 1 | Large Null Invariant Error | 1.127e-9 | NI=1.127e-9 (integration accuracy issue) |
| (841, 571) | 0 | Large Null Invariant Error | 3.285e-9 | NI=3.285e-9 (integration accuracy issue) |
| (847, 151) | 0 | Large Null Invariant Error | 1.005e-9 | NI=1.005e-9 (integration accuracy issue) |
| (870, 670) | 0 | Large Null Invariant Error | 1.396e-9 | NI=1.396e-9 (integration accuracy issue) |
| (879, 370) | 0 | Large Null Invariant Error | 1.089e-9 | NI=1.089e-9 (integration accuracy issue) |
| (879, 370) | 1 | Large Null Invariant Error | 1.089e-9 | NI=1.089e-9 (integration accuracy issue) |
| (941, 913) | 0 | Large Null Invariant Error | 1.062e-9 | NI=1.062e-9 (integration accuracy issue) |
| (971, 857) | 0 | Large Null Invariant Error | 1.062e-9 | NI=1.062e-9 (integration accuracy issue) |
| (971, 857) | 1 | Large Null Invariant Error | 1.062e-9 | NI=1.062e-9 (integration accuracy issue) |
| (1000, 917) | 0 | Large Null Invariant Error | 1.076e-9 | NI=1.076e-9 (integration accuracy issue) |
| (1000, 917) | 1 | Large Null Invariant Error | 1.076e-9 | NI=1.076e-9 (integration accuracy issue) |
| (1025, 241) | 1 | Large Null Invariant Error | 1.329e-9 | NI=1.329e-9 (integration accuracy issue) |
| Pixel | Order | Category | Value | Details |
|---|---|---|---|---|
| (523, 609) | 0 | Spatial Discontinuity | 9.530e0 | NI differs from neighbors (local MAD z=9.5, Δ=9.9%) |
| (524, 484) | 0 | Spatial Discontinuity | 1.642e1 | NI differs from neighbors (local MAD z=16.4, Δ=10.1%) |
| (524, 512) | 0 | Spatial Discontinuity | 9.668e0 | NI differs from neighbors (local MAD z=9.7, Δ=7.8%) |
| (526, 639) | 0 | Spatial Discontinuity | 6.679e0 | NI differs from neighbors (local MAD z=6.7, Δ=8.9%) |
| (532, 595) | 0 | Spatial Discontinuity | 6.357e0 | NI differs from neighbors (local MAD z=6.4, Δ=13.7%) |
| (538, 388) | 0 | Spatial Discontinuity | 1.450e1 | NI differs from neighbors (local MAD z=14.5, Δ=10.8%) |
| (538, 414) | 0 | Spatial Discontinuity | 2.762e1 | NI differs from neighbors (local MAD z=27.6, Δ=15.9%) |
| (539, 414) | 0 | Spatial Discontinuity | 1.897e1 | NI differs from neighbors (local MAD z=19.0, Δ=11.0%) |
| (540, 413) | 0 | Spatial Discontinuity | 7.675e0 | NI differs from neighbors (local MAD z=7.7, Δ=20.1%) |
| (544, 419) | 0 | Spatial Discontinuity | 5.646e1 | NI differs from neighbors (local MAD z=56.5, Δ=25.1%) |
| (551, 372) | 0 | Spatial Discontinuity | 9.645e0 | NI differs from neighbors (local MAD z=9.6, Δ=11.6%) |
| (551, 616) | 0 | Spatial Discontinuity | 1.220e1 | NI differs from neighbors (local MAD z=12.2, Δ=9.5%) |
| (552, 707) | 0 | Spatial Discontinuity | 6.302e0 | NI differs from neighbors (local MAD z=6.3, Δ=19.5%) |
| (554, 730) | 0 | Spatial Discontinuity | 6.416e0 | NI differs from neighbors (local MAD z=6.4, Δ=5.6%) |
| (557, 533) | 0 | Spatial Discontinuity | 1.215e1 | NI differs from neighbors (local MAD z=12.2, Δ=16.7%) |
| (560, 460) | 0 | Spatial Discontinuity | 1.378e1 | NI differs from neighbors (local MAD z=13.8, Δ=13.0%) |
| (563, 358) | 0 | Spatial Discontinuity | 2.249e1 | NI differs from neighbors (local MAD z=22.5, Δ=10.5%) |
| (563, 703) | 0 | Spatial Discontinuity | 1.064e2 | NI differs from neighbors (local MAD z=106.4, Δ=12.1%) |
| (564, 670) | 0 | Spatial Discontinuity | 8.512e0 | NI differs from neighbors (local MAD z=8.5, Δ=11.7%) |
| (577, 720) | 0 | Spatial Discontinuity | 7.355e1 | NI differs from neighbors (local MAD z=73.5, Δ=11.8%) |
| Pixel | Order | Category | Value | Details |
|---|---|---|---|---|
| (640, 544) | 0 | Extreme φ-Wraps | 9.584e-1 | φ-wraps=0.96 > p99=0.96 |
| (640, 547) | 0 | Extreme φ-Wraps | 9.582e-1 | φ-wraps=0.96 > p99=0.96 |
| (643, 563) | 0 | Extreme φ-Wraps | 9.583e-1 | φ-wraps=0.96 > p99=0.96 |
| (643, 567) | 0 | Extreme φ-Wraps | 9.577e-1 | φ-wraps=0.96 > p99=0.96 |
| (643, 571) | 0 | Extreme φ-Wraps | 9.571e-1 | φ-wraps=0.96 > p99=0.96 |
| (644, 541) | 0 | Extreme φ-Wraps | 9.609e-1 | φ-wraps=0.96 > p99=0.96 |
| (645, 541) | 0 | Extreme φ-Wraps | 9.615e-1 | φ-wraps=0.96 > p99=0.96 |
| (645, 544) | 0 | Extreme φ-Wraps | 9.613e-1 | φ-wraps=0.96 > p99=0.96 |
| (645, 571) | 0 | Extreme φ-Wraps | 9.583e-1 | φ-wraps=0.96 > p99=0.96 |
| (646, 560) | 0 | Extreme φ-Wraps | 9.604e-1 | φ-wraps=0.96 > p99=0.96 |
| (646, 577) | 0 | Extreme φ-Wraps | 9.578e-1 | φ-wraps=0.96 > p99=0.96 |
| (648, 566) | 0 | Extreme φ-Wraps | 9.608e-1 | φ-wraps=0.96 > p99=0.96 |
| (650, 548) | 0 | Extreme φ-Wraps | 9.640e-1 | φ-wraps=0.96 > p99=0.96 |
| (650, 552) | 0 | Extreme φ-Wraps | 9.637e-1 | φ-wraps=0.96 > p99=0.96 |
| (650, 554) | 0 | Extreme φ-Wraps | 9.635e-1 | φ-wraps=0.96 > p99=0.96 |
| (650, 581) | 0 | Extreme φ-Wraps | 9.593e-1 | φ-wraps=0.96 > p99=0.96 |
| (650, 593) | 0 | Extreme φ-Wraps | 9.567e-1 | φ-wraps=0.96 > p99=0.96 |
| (651, 546) | 0 | Extreme φ-Wraps | 9.648e-1 | φ-wraps=0.96 > p99=0.96 |
| (651, 555) | 0 | Extreme φ-Wraps | 9.640e-1 | φ-wraps=0.96 > p99=0.96 |
| (651, 569) | 0 | Extreme φ-Wraps | 9.621e-1 | φ-wraps=0.96 > p99=0.96 |
| Pixel | Order | Category | Value | Details |
|---|---|---|---|---|
| (509, 523) | 0 | Long Affine Parameter (Outer Disc) | 8.391e1 | λ=83.9M (MAD z=4.2, outer disc edge) |
| (509, 530) | 0 | Long Affine Parameter (Outer Disc) | 8.391e1 | λ=83.9M (MAD z=4.2, outer disc edge) |
| (509, 532) | 0 | Long Affine Parameter (Outer Disc) | 8.391e1 | λ=83.9M (MAD z=4.2, outer disc edge) |
| (509, 543) | 0 | Long Affine Parameter (Outer Disc) | 8.391e1 | λ=83.9M (MAD z=4.2, outer disc edge) |
| (509, 544) | 0 | Long Affine Parameter (Outer Disc) | 8.391e1 | λ=83.9M (MAD z=4.2, outer disc edge) |
| (509, 552) | 0 | Long Affine Parameter (Outer Disc) | 8.392e1 | λ=83.9M (MAD z=4.2, outer disc edge) |
| (509, 556) | 0 | Long Affine Parameter (Outer Disc) | 8.392e1 | λ=83.9M (MAD z=4.2, outer disc edge) |
| (510, 502) | 0 | Long Affine Parameter (Outer Disc) | 8.388e1 | λ=83.9M (MAD z=4.2, outer disc edge) |
| (510, 508) | 0 | Long Affine Parameter (Outer Disc) | 8.387e1 | λ=83.9M (MAD z=4.2, outer disc edge) |
| (510, 518) | 0 | Long Affine Parameter (Outer Disc) | 8.386e1 | λ=83.9M (MAD z=4.2, outer disc edge) |
| (510, 528) | 0 | Long Affine Parameter (Outer Disc) | 8.386e1 | λ=83.9M (MAD z=4.2, outer disc edge) |
| (510, 545) | 0 | Long Affine Parameter (Outer Disc) | 8.386e1 | λ=83.9M (MAD z=4.2, outer disc edge) |
| (510, 553) | 0 | Long Affine Parameter (Outer Disc) | 8.387e1 | λ=83.9M (MAD z=4.2, outer disc edge) |
| (510, 558) | 0 | Long Affine Parameter (Outer Disc) | 8.388e1 | λ=83.9M (MAD z=4.2, outer disc edge) |
| (510, 560) | 0 | Long Affine Parameter (Outer Disc) | 8.388e1 | λ=83.9M (MAD z=4.2, outer disc edge) |
| (510, 565) | 0 | Long Affine Parameter (Outer Disc) | 8.389e1 | λ=83.9M (MAD z=4.2, outer disc edge) |
| (510, 566) | 0 | Long Affine Parameter (Outer Disc) | 8.389e1 | λ=83.9M (MAD z=4.2, outer disc edge) |
| (510, 568) | 0 | Long Affine Parameter (Outer Disc) | 8.390e1 | λ=83.9M (MAD z=4.2, outer disc edge) |
| (510, 570) | 0 | Long Affine Parameter (Outer Disc) | 8.390e1 | λ=83.9M (MAD z=4.2, outer disc edge) |
| (511, 496) | 0 | Long Affine Parameter (Outer Disc) | 8.385e1 | λ=83.8M (MAD z=4.2, outer disc edge) |
Downsampled heatmap showing where outliers are concentrated. Darker = more outliers in region, color = highest severity level.
This report analyzes high-precision f64 geodesic data from Kerr spacetime ray tracing. Each pixel traces a null geodesic (light path) backward from the camera through curved spacetime, potentially crossing the accretion disc multiple times due to extreme gravitational lensing.
The null invariant (NI) measures numerical accuracy. Values <1e-9 indicate excellent geodesic integration.